The Annals of Applied Probability

Social contact processes and the partner model

Eric Foxall, Roderick Edwards, and P. van den Driessche

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Abstract

We consider a stochastic model of infection spread on the complete graph on $N$ vertices incorporating dynamic partnerships, which we assume to be monogamous. This can be seen as a variation on the contact process in which some form of edge dynamics determines the set of contacts at each moment in time. We identify a basic reproduction number $R_{0}$ with the property that if $R_{0}<1$ the infection dies out by time $O(\log N)$, while if $R_{0}>1$ the infection survives for an amount of time $e^{\gamma N}$ for some $\gamma>0$ and hovers around a uniquely determined metastable proportion of infectious individuals. The proof in both cases relies on comparison to a set of mean-field equations when the infection is widespread, and to a branching process when the infection is sparse.

Article information

Source
Ann. Appl. Probab., Volume 26, Number 3 (2016), 1297-1328.

Dates
Received: December 2014
Revised: April 2015
First available in Project Euclid: 14 June 2016

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1465905004

Digital Object Identifier
doi:10.1214/15-AAP1117

Mathematical Reviews number (MathSciNet)
MR3513591

Zentralblatt MATH identifier
1345.60115

Subjects
Primary: 60J25: Continuous-time Markov processes on general state spaces
Secondary: 92B99: None of the above, but in this section

Keywords
SIS model contact process interacting particle systems

Citation

Foxall, Eric; Edwards, Roderick; van den Driessche, P. Social contact processes and the partner model. Ann. Appl. Probab. 26 (2016), no. 3, 1297--1328. doi:10.1214/15-AAP1117. https://projecteuclid.org/euclid.aoap/1465905004


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References

  • [1] Bezuidenhout, C. and Grimmett, G. (1990). The critical contact process dies out. Ann. Probab. 18 1462–1482.
  • [2] Broman, E. I. (2007). Stochastic domination for a hidden Markov chain with applications to the contact process in a randomly evolving environment. Ann. Probab. 35 2263–2293.
  • [3] Chatterjee, S. and Durrett, R. (2009). Contact processes on random graphs with power law degree distributions have critical value 0. Ann. Probab. 37 2332–2356.
  • [4] Durrett, R. (1980). On the growth of one-dimensional contact processes. Ann. Probab. 8 890–907.
  • [5] Durrett, R. (2010). Probability: Theory and Examples, 4th ed. Cambridge Univ. Press, Cambridge.
  • [6] Harris, T. E. (1978). Additive set-valued Markov processes and graphical methods. Ann. Probab. 6 355–378.
  • [7] Harris, T. E. (2002). The Theory of Branching Processes. Dover, Mineola, NY.
  • [8] Liggett, T. M. (1985). Interacting Particle Systems. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 276. Springer, New York.
  • [9] Liggett, T. M. (1999). Stochastic Interacting Systems: Contact, Voter and Exclusion Processes. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 324. Springer, Berlin.
  • [10] Mountford, T., Valesin, D. and Yao, Q. (2013). Metastable densities for the contact process on power law random graphs. Electron. J. Probab. 18 1–36.
  • [11] Pemantle, R. (1992). The contact process on trees. Ann. Probab. 20 2089–2116.
  • [12] Peterson, J. (2011). The contact process on the complete graph with random vertex-dependent infection rates. Stochastic Process. Appl. 121 609–629.
  • [13] Remenik, D. (2008). The contact process in a dynamic random environment. Ann. Appl. Probab. 18 2392–2420.
  • [14] van den Driessche, P. and Watmough, J. (2002). Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission. Math. Biosci. 180 29–48.